66 / C6-Q: Ancient Defect Rigidity, Local Growth Lift, Strain-Projection Tail Reduction, and Spatial Carrier Rebinding

The final difficulty C6-P left behind seemed to be the nonlocal tail of TS's operator channel $P_{st}$. C6-Q discovers a major correction: for the temporal $H^1$ strain-growth ledger that TS actually uses, $P_{st}$ is simply not an intrinsic spatial-carrier object at all. Using Miller's exact identity $\langle-\Delta S,\omega\otimes\omega\rangle=0$ together with $-\Delta S\in L^2_{st}$, $P_{st}$ can be removed entirely from the growth pairing; C6-Q.1 Projection-Free $H^1$ Growth Identity therefore proves that $E_1'=-\nu\|\Delta S\|_2^2-\langle(u\cdot\nabla)S+S^2,-\Delta S\rangle$ has a fully local exact spatial-density representative $g_O^{loc}$ — containing no $P_{st}$ and no singular pressure integral. C6-Q.2 Operator-Lift Nonuniqueness Guard corrects the "canonical" wording used back in C6-E: a temporal operator-growth margin does not determine a unique positive spatial lift, and the provenance of the Projected Representative Lift and the Local Growth Representative Lift must be labeled separately. C6-Q.3 Projected-Tail Deletion for the TS Growth Carrier formally declares that the nonlocal $P_{st}$ tail is not an intrinsic obstruction to localizing the TS growth carrier — but it draws a clear boundary: this only removes the carrier-tail problem, not every use of $P_{st}$, because Miller's genuine critical-operator criterion still depends on the norm of $Q_{SV}=P_{st}(\cdots)$ itself, so growth carrier ≠ operator-norm carrier. Even for the operator-norm channel that genuinely needs $P_{st}$, C6-Q.4 Explicit Strain-Projection Formula derives directly, from the Miller–Sawyer isometric isomorphism $T^*T=\frac12I$, the explicit formula $P_{st}M=-2\nabla_{\rm sym}(-\Delta)^{-1}P_{df}\operatorname{div}M$, proving that $P_{st}$ is an order-zero Calderón–Zygmund matrix operator with kernel $K_{st}(x)=|x|^{-3}\Omega(x/|x|)$; C6-Q.5/6 Far-Tail Oscillation Lemma / Local-Constant Tail Reduction prove that, for a bounded source, the far-field tail beyond a fixed core amounts to at most "a constant matrix mode plus $O(R_0/R)$ vanishing oscillation" — far-field nonlocality is compressed down to a finite dimension. C6-Q.8 Conditional Ancient TS Local-Growth Inheritance therefore proves that, under five conditions, $TS_{growth}^{anc}$ can be carried over to the ancient limit with no far-tail assumption on $P_{st}$ whatsoever — explicitly flagged as a major correction to C6-P. The round's second genuinely positive breakthrough handles the spatial-carrier escape left by C6-P: C6-Q.9 Satellite Ancient Defect Extraction proves that spatial escape with $|z_n|\to\infty$ is not the end of the road — by recentering via translation symmetry, any nonzero local absolute defect load can have a "Satellite Ancient Defect Profile" carrying the same defect extracted from it; all three labels HF, GP, and TS each acquire their own satellite version, and C6-Q.10 Satellite Physical-Center Trichotomy further classifies the physical-distance relationship between the satellite center and the original recorded peak. Going further, if an ancient defect recurs repeatedly as $\tau_j\to-\infty$, C6-Q.11 Ancient Recurrence → Eternal Profile Extraction proves that a spacetime translation can extract a bounded eternal N–S solution defined over all of time $\mathbb R$, carrying the same nonzero local defect — honestly stating that this is not a contradiction, since the general 3D bounded ancient/eternal Liouville problem remains open, but rather a still-stronger compactification reduction. The round gathers everything into C6-Q.12 Ancient Defect Reduction: $GP_{\rm loc}^{anc}\vee HF_{k,\rm geom}^{anc}\vee TS_{growth}^{anc}\vee OP_{\rm norm}^{anc}\vee$ Order-Geometry Escape $\vee$ Eternal Defect Profile $\vee$ FLAT — FLAT carries no defect at all, and spatial-carrier escape is no longer an independent terminal classification. Formally hands off to C6-R — Eternal Defect Profiles, Analytic Order-Geometry Rigidity, and Operator-Constant-Mode Closure, with ten constituent obligations directly attacking: whether nontrivial bounded eternal TS/GP/HF defect states exist, and whether order-geometry escape can truly survive under a uniform analytic baseline.